Question

Random sampling and confidence interval problem (screenshots attached)

Answer 1

Here is the example that they referred to in the problem:

Answer 2

I attempted to follow the example problem, but I guess the answers that I came up with weren't correct!

Answer 3

what is the problem

Answer 4

Were you stuck right from A?

Answer 5

So basically A) We need to construct the limits for our 95% CI..using the table we have, knowing we have 2n = (2 * 15) = 30 degrees of freedom...we arrive at 46.979 for upper area and 16.791 for lower area

Giving us *based off your example

\[\large P(16.791 < 2\lambda \sum x_i < 46.979) = 0.95\]

Lets just go ahead and say that \(\large \sum x_i = 63.3\)

So we divide everything by \(\large 2\sum x_i\) or 126.6

\[\large (\frac{16.791}{126.6}) < \lambda < \frac{46.979}{126.6})\]

The expected value apparently means we need the reciprocal of everything, and switch the limits

\[\large \frac{126.6}{46.979} < \frac{1}{\lambda} < \frac{126.6}{16.791}\]

Giving us

\(\large (2.69 , 7.54)\) as our interval

Answer 6

@johnweldon1993 Hmm, maybe I made a small algebra error because I followed the same process! X)

How would you go about c?

Answer 7

Hmm from what I'm reading * just general googling x) *

We would find the standard deviation of an exponential random variable to be \(\large \frac{1}{\lambda}\) But that can't be right because that is the expected value as well *what we solved for

Answer 8

So if you have any insight, please share haha

Answer 9

To be honest, I've tried researching it! My professor actually never went over it in his lectures or his notes (probably why he included the example). We never even touched this part of the material of chi-squared distribution. How frustrating!

I shall consult the textbook and see if I can find anything else

Answer 10

Extremely! I finished Stats last year so my memory has faded, which doesn't help the cause

Answer 11

I can do physics, calculus, and differential equations; but statistics has been quite the road bump for me.

Answer 12

^Basically my mantra that semester lol

Answer 13

My source:

https://math.berkeley.edu/~scanlon/m16bs04/ln/16b2lec31.pdf

Basically, the integration makes sense, and it is pointing to the fact that the expected value = standard deviation

Answer 14

Just first and second page there...

Answer 15

But yeah, Stats and Fourier Series...my two greatest enemies in terms of mathematics

Answer 16

Interesting, but it worked! They are in fact the same.

Haven't gone over Fourier series, but I probably will eventually. I'm majoring in Mechanical Engineering (actually about to switch to Aerospace). I hate series at first, but after working with them for a little bit I get used to them again until I forget them the following semester X)

Answer 17

Ahh a fellow ME! Wonderful! lol, I've also considered Aerospace, but idk, I feel ME is just more broad and if I want to expand further into another branch I would have that option

Answer 18

Hehe. I've been hearing from professionals that they're so interchangeable. But my goal is to intern/work for an aerospace company (I live an hour away from NASA KSC/SpaceX), so I figured the switch to learn the proper material was necessary.

Answer 19

:O Definitely a good career-orientated choice then! Where I live the most noteworthy company is Pratt & Whitney *Which I plan on interning this coming summer* Not too much up here in the NorthEast for Defense / Aerospace Engineering

Answer 20

SpaceX would be beautiful but I heard that applications are competitive and they're very selective (GPA minimum 3.5 while norm is 3.0). Pratty & Whitney is great too, especially for mechanical engineers! They also provide a lot for Lockheed Martin, which I also live 30 minutes away from! Planning on submitting an application there soon, too. I think the hardest part is getting the internship, because if you graduate without any internship experience in the engineering field, you're at a big disadvantage. Once you get the internship, most of the time the companies will keep you for full employment.

Answer 21

Sad but hard truth, I have had MANY friends graduate and still have trouble finding work because of the lack of experience.

Well, come on now, Engineering is one hell of a competitive field lol, applications are going to be cutthroat, just like some of the interviews...haha the best phone interview I had left me hanging up like...what did I just say? They throw you for a loop big time to see how you respond under pressure and see if you can act on your feet!

I have a 4.0 right now, which I'm sure will fall sadly this semester *Mechatronics is kicking my butt* The programming aspect...just too much to take all in one semester* But I plan to apply before those grades go in and hope they see it as...oh...4.0 huh? XD

Answer 22

Yeah! Especially since more and more students are entering STEM related majors, making the job positions harder to get!

I haven't had any engineering internships or interviews yet, which scares me! But I know it needs to be done X)

4.0 is sick! My test averages for most courses ends up being in the B range, which, no matter how high your homework or quizzes are, ultimately still pulls the rest of your grade down to the B range... So I've been trying to boost my ~3.5 a little bit